De-equivariantization of Hopf algebras

arXiv:1206.0410v1 [math.QA] 2 Jun 2012

DE-EQUIVARIANTIZATION OF HOPF ALGEBRAS ´ ANGIONO, CESAR ´ IVAN GALINDO AND MARIANA PEREIRA Abstract. We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras A(H, G, Φ) attached to a coradically-graded pointed Hopf algebra H and some extra data.

Introduction Actions of groups over abelian categories have been studied in recent years with the purpose of constructing, describing and studying categories with symmetries. For example, Gaitsgory [G] introduced the notion of the action of an affine group scheme G over a C-linear abelian category C and the category of G-equivariant objects C G , called the equivariantization of C by G. The category C G has an action of Rep(G) and the category of Hecke eigen-objects in C G is again C. In general, if Rep(G) acts on an abelian category C, then the category of Hecke eigen-objects in C is called the de-equivariantization of C by G. Equivariantization and de-equivariantization are standard techniques in theory of fusion categories [DGNO] and have been applied in geometric Langlands program [FG] and quantum groups [ArG]. Now, if C is a tensor category and the action of Rep(G) over C is tensorial, then the de-equivariantization has a natural tensor structure. A special but very important type of tensor categories are those equivalent to the category of corepresentations of a Hopf algebra, which include representations of algebraic groups, quantum groups, compact groups, etc. If C is the category of comodules (or finite dimensional modules) over a Hopf algebra, then C G → C → Vec is a fiber functor on C G (where C G → C is the forgetful functor) and by Tannakian duality C G is the category of comodules over Hopf algebra. Thus, the family of Hopf algebras is closed under equivariantization, in the sense that we obtain new categories which are equivalent to categories of corepresentations of Hopf algebras. This is not the case for the de-equivariantization process since the de-equivariantization of comodules over a Hopf algebra is not always equivalent to the category of corepresentations over a Hopf algebra (see Subsection 3.3, for concrete examples). However, under some mild conditions, it is always the category of corepresentations over a coquasibialgebra. As a consequence there exist coquasi-Hopf algebras not twist equivalent to Hopf algebras, which admit an equivariantization equivalent to a Hopf algebra. This phenomenon was used in [EG] to relate the Drinfeld doubles of some quasiHopf algebras with small quantum groups, and in [A1] in order to classify the family 2010 Mathematics Subject Classification. 16W30, 18D10, 19D23. The work of I. A. was partially supported by CONICET, FONCyT-ANPCyT and Secyt (UNC). M.P. is grateful for the support from the grant ANII FCE 2007-059. 1

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ANGIONO, GALINDO AND PEREIRA

of basic quasi-Hopf algebras with cyclic group of one-dimensional representations, under some mild conditions. In this paper we study the de-equivariantization of the category of comodules over a Hopf algebra by an affine group scheme and apply Tannakian techniques to realize the de-equivariantization as the tensor category of comodules over a coquasibialgebra. We apply the construction to interpret the central extensions of Hopf algebras as a particular example, and an additional application to the context of pointed finite tensor categories, extending the family of examples obtained in [EG], [Ge], [A1]. The organization of the paper is the following. In Section 1 we recall the definitions related with the main construction of this paper. First, the relation between affine group schemes and commutative algebras, then co-quasi bialgebras, and finally the center of a tensor category. In Section 2 we build a co-quasi Hopf algebra which represents the tensor category obtained as the de-equivariantization of the category of co-representations of a Hopf algebra. To do this, we consider central braided Hopf bialgebras, which are in correspondence with inclusions of tensor categories of comodules over Hopf algebras with certain factorization through the center, making emphasis on the case of algebras of functions over an affine group (in particular, over finite groups). We then obtain the corresponding coquasi-Hopf algebra representing a de-equivariantization over the comodules of a Hopf algebra by a Tannakian reconstruction. Finally, Section 3 contains some applications of the previous results. The main one is the case of finite-dimensional pointed Hopf algebras, which gives place to a general construction of pointed coquasi-Hopf algebras, and consequently finite pointed tensor categories.

1. Preliminaries In this section we recall some definitions and results on Hopf algebras, affine group schemes and coquasi-Hopf algebras. For further reading on these topics we the reader to [M], [W] and [S1] respectively. Throughout the paper we work over an arbitrary field k. Algebras and coalgebras are always defined over k. For a coalgebra (C, ∆, ε) we shall use Sweedler’s notation omitting the sum symbol, that is ∆(c) = c1 ⊗ c2 for all c ∈ C. Similarly if (M, λ) is a left C-comodule, then λ(m) = m−1 ⊗ m0 ∈ C ⊗ M for all m ∈ M . The category of left C-comodules shall be denoted by C M. 1.1. Affine group scheme and commutative Hopf algebras. Let k-Alg denote the category of commutative k-algebras and Grp the category of groups. An affine group scheme over k is a representable functor G : k-Alg → Grp. By Yoneda’s lemma the commutative algebra that represents G is unique up to isomorphisms, and we shall denote it by O(G). The group structures on G(A), A ∈ k-Alg, determine natural transformations m : G × G → G, 1 : Sp(k) → G, i : G → G,

DE-EQUIVARIANTIZATION OF HOPF ALGEBRAS

3

and they define algebra maps ∆ : O(G) → O(G) ⊗ O(G), ε : O(G) → k, S : O(G) → O(G), that give a Hopf algebra structure on O(G). Conversely, if K is a commutative Hopf algebra, then Spec(K) : k-Alg → Set, A 7→ Alg(K, A) is an affine group scheme with group structure given by the convolution product and this defines an antiequivalence of categories between affine groups schemes over k and commutative Hopf algebras over k. Under this equivalence the category of representations of G is equivalent to the category of O(G)-comodules, and quasi-coherent sheaves on G are O(G)-modules. 1.2. Coquasi-bialgebras. A coquasi-bialgebra (H, m, u, ω, ∆, ε) is a coalgebra (H, ∆, ε) together with coalgebra morphisms: • the multiplication m : H ⊗ H −→ H (denoted m(h ⊗ g) = hg), • the unit u : k −→ H (where we call u(1) = 1H ), and a convolution invertible element ω ∈ (H⊗H⊗H)∗ such that for all h, g, k, l ∈ H: (1.1)

h1 (g1 k1 )ω(h2 , g2 , k2 ) =

(1.2) (1.3)

1H h = ω(h1 g1 , k1 , l1 )ω(h2 , g2 , k2 l2 ) =

(1.4)

ω(h, 1H , g) =

ω(h1 , g1 , k1 )(h2 g2 )k2 h1H = h ω(h1 , g1 , k1 ) ω(h2 , g2 k2 , l1 )ω(g3 , k3 , l2 ) ε(h)ε(g).

Note that ω(1H , h, g) = ω(h, g, 1H ) = ε(h)ε(g) for each g, h ∈ H. A coquasi-Hopf algebra is a coquasi-bialgebra H endowed with a coalgebra antihomomorphism S : H −→ H (the antipode) and elements α, β ∈ H ∗ satisfying, for all h ∈ H: (1.5) (1.6)

S(h1 )α(h2 )h3 h1 β(h2 )S(h3 )

(1.7)

ε(h)

= α(h)1H = β(h)1H = ω(h1 β(h2 ), S(h3 ), α(h4 )h5 ) = ω −1 (S(h1 ), α(h2 )h3 β(h4 ), S(h5 )).

The category of left H-comodules H M is rigid and monoidal, where the tensor product is over the base field and the comodule structure of the tensor product is the codiagonal one. The associator is given by φU,V,W φU,V,W ((u ⊗ v) ⊗ w)

:

(U ⊗ V ) ⊗ W −→ U ⊗ (V ⊗ W )

= ω(u−1 , v−1 , w−1 )u0 ⊗ (v0 ⊗ w0 )

for u ∈ U , v ∈ V , w ∈ W and U, V, W ∈ H M. The dual coactions are given by S and S −1 , as in the case of Hopf algebras.

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1.3. The center construction and the category of Yetter-Drinfeld modules. The center construction produces a braided monoidal category Z(C) from any monoidal category C, see [K]. The objects of Z(C) are pairs (Y, c−,Y ), where Y ∈ C and cX,Y : X ⊗ Y → Y ⊗ X are isomorphisms natural in X satisfying cX⊗Y,Z = (cXZ ⊗ id Y )(id X ⊗ cY,Z ) and cI,Y = id Y , for all X, Y, Z ∈ C. The braided monoidal structure is given in the following way: • the tensor product is (Y, c−,Y ) ⊗ (Z, c−,Z ) = (Y ⊗ Z, c−,Y ⊗Z ), where cX,Y ⊗Z = (id Y ⊗ cX,Z )(cX,Y ⊗ id Z ) : X ⊗ Y ⊗ Z → Y ⊗ Z ⊗ X, for all X ∈ C, • the identity element is (I, c−,I ), cZ,I = id Z • the braiding is the morphism cX,Y . Let H be a Hopf algebra with bijective antipode. We shall denote by H M the tensor category of left H-comodules. The category Z(H M) is braided equivalent to the category H H YD of left-left Yetter-Drinfeld modules, whose objects are left H-comodules and left H-modules M satisfying the condition (1.8)

(h1 ⇀ m)−1 h2 ⊗ (h1 ⇀ m)0 = h1 m−1 ⊗ h2 ⇀ m0

for all m ∈ M , h ∈ H. A Yetter-Drinfeld module N becomes an object in Z(H M) by cM,N (m ⊗ n) = m−1 ⇀ n ⊗ m0 , −1 and inverse cM,N (n ⊗ m) = m0 ⊗ S −1 (m1 ) ⇀ n. 2. De-equivariantization of Hopf algebras 2.1. Central inclusion and braided central Hopf subalgebras. Let H be a Hopf algebra with bijective antipode. Let G be an affine group scheme over k and O(G) the Hopf algebra of regular functions over G. A central inclusion of G in H is a braided monoidal inclusion ι : Rep(G) ֒→ H Z(H M) ∼ = H H YD, such that the braiding of H YD restricts to the usual symmetric H braiding of Rep(G), and the composition Rep(G) ֒→ H M gives an H YD → inclusion. In order to describe in Hopf-theoretical terms the central inclusions, we need the following concept. Definition 2.1. Let H be a Hopf algebra. A braided central Hopf subalgebra of H is a pair (K, r), where K ⊂ H is a Hopf subalgebra, and r : H ⊗ K → k is a bilinear form such that: (2.1)

r(hh′ , k) = r(h′ , k1 )r(h, k2 ),

(2.2)

r(h, kk ′ ) = r(h1 , k)r(h2 , k ′ ),

(2.3)

r(h, 1) = ε(h),

r(1, k) = ε(k),

(2.4)

r(h1 , k1 )k2 h2 = h1 k1 r(h2 , k2 ),

(2.5)

r(k, k ′ ) = ε(kk ′ ),

for all k, k ′ ∈ K, h, h′ ∈ H. Remark 2.2. (1) The conditions (2.1), (2.2), (2.3), say that r : H ⊗ K → k is a Hopf skew pairing, so in particular r has a convolution-inverse r−1 (h, k) = r(h, S(k)),

(h ∈ H, k ∈ K).


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(2) The algebra K is commutative by (2.4) and (2.5). (3) For all V ∈ H M, W ∈ K M, the map r defines a natural isomorphism cV,W : V ⊗ W → W ⊗ V, v ⊗ w 7→ r(v−1 , w−1 )w0 ⊗ v0 in H M, and these isomorphisms define a braided inclusion K M → Z(H M) = H H YD. (4) The condition (2.5) implies that K is a commutative algebra in Z(H M). For example, any central Hopf subalgebra K ⊂ H is braided central with r = εH ⊗ εK . Conversely, if K ⊂ H is a braided central Hopf subalgebra with r = εH ⊗ εK then K is a central Hopf subalgebra. Lemma 2.3. Let K ⊂ H be a braided central Hopf subalgebra. Then (2.6)

r(xh, k) = r(hx, k) = ε(x)r(h, k),

for all x, k ∈ K, h ∈ H, Proof. It follows from conditions (2.1) and (2.5).

The following result exhibits the relevance of braided central Hopf subalgebras. Theorem 2.4. Let H be a Hopf algebra and K ⊂ H a commutative Hopf subalgebra. Then the following set of data are equivalent: (1) A map r : H ⊗ K → k such that (K, r) is a braided central Hopf subalgebra of H. (2) A braided monoidal functor F : K M → Z(H M) = H H YD such that the H composition with the forgetful functor Z(H M) = H M is an H YD → inclusion. (3) A Hopf algebra map γ : K → (H ◦ )cop with γ(k)|K = ε and hγ(k1 ), h1 ik2 h2 = h1 k1 hγ(k2 ), h2 i for all h ∈ H, k ∈ K ( H ◦ denotes the finite dual Hopf algebra). Proof. (1) ⇒ (2) Let M ∈ K M, then the map ⇀: H ⊗ M → M, h ⊗ m 7→ r(h, m−1 )m0 , defines a structure of H-module, that satisfies the Yetter-Drinfeld compatibility by (2.4). (2) ⇒ (3) Since every comodule is a colimit of finite dimensional comodules, the image of the monoidal functor K M → H H YD → H M lives in the tensor subcategory of H M of H-modules that are colimits of finite dimensional H-modules, then the ∼ (H ◦ )cop M induces a unique Hopf monoidal functor K M → H H YD → H M = ◦ cop algebra map γ : K → (H ) given by h ⇀ m = hγ(m−1 ), him0 , for all h ∈ H, m ∈ M and M ∈ It is enough to prove that

K

M.

h1 m−2 hγ(m−1 ), h2 i ⊗ m0 = m−1 h2 hγ(m−2 ), h1 i ⊗ m0 , for all m ∈ M , M ∈

K

M. Indeed, (1.8) implies that

h1 m−2 hγ(m−1 , h2 i ⊗ m0 = h1 m−2 ⊗ hγ(m−1 , h2 im0 = h1 m−2 ⊗ h2 ⇀ m0 = (h1 ⇀ m)−1 h2 ⊗ (h1 ⇀ m)0 = m−1 h2 ⊗ hγ(m−2 , h1 im0 = m−1 h2 hγ(m−2 ), h1 i ⊗ m0 .

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(3) ⇒ (1) The map r(h, k) = hγ(k), hi defines a braided central structure over K. The following result in an immediate consequence of Theorem 2.4. Corollary 2.5. Let H be a Hopf algebra. There exist a bijective correspondence between central inclusions of G in H and braided central Hopf subalgebras K of H such that K ∼ = O(G) as Hopf algebras. 2.2. De-equivariantization of a Hopf algebra by an affine group scheme. Let H be a Hopf algebra and G be an affine group scheme. Let K ⊂ H a braided central Hopf subalgebra with K = O(G). The algebra O(G) is a commutative algebra in the symmetric category Rep(G), and thus a commutative algebra in the braided tensor category H H YD (see Remark 2.2 item (4)). Therefore, the algebra O(G) is braided commutative. We define the de-equivariantization H M(G) of H M by G, as the category of H-equivariant sheaves on G, that is the category of left O(G)-modules in H M. H Now, the category H M is a tensor category with G MG of O(G)-bimodules in the tensor product M ⊗O(G) N . We shall see in the next proposition that this tensor product induces a monoidal structure on H M(G). Proposition 2.6. Let V ∈ H M(G) with left O(G)-module structure ⇀: O(G) ⊗ V → V and left H-comodule structure λ : V → O(G) ⊗ V, v 7→ v−1 ⊗ v0 . The map ↼: V ⊗ O(G) → V, v ↼ x = r(v−1 , x1 )x2 ⇀ v0 , makes V an object in H G MG . This rule defines a fully faithful strict monoidal functor from H M(G) to H G MG . Proof. Let V ∈ H M(G) with left O(G)-module structure ⇀: O(G) ⊗ V → V and left H-comodule structure λ : V → O(G) ⊗ V, v 7→ v−1 ⊗ v0 . (1) The map ↼: V ⊗ O(G) → V defines a right O(G)-module structure: for any v ∈ V and x, y ∈ O(G), (v ↼ x) ↼ y = (r(v−1 , x1 )x2 ⇀ v0 ) ↼ y = r(v−1 , x1 )r((x2 ⇀ v0 )−1 , y1 )y2 ⇀ (x2 ⇀ v0 )0 = r(v−2 , x1 )r(x2 v−1 , y1 )y2 ⇀ (x3 ⇀ v0 ) = r(v−2 , x1 )r(v−1 , y1 )y2 x2 ⇀ v0 = r(v−1 , y1 x1 )y2 x2 ⇀ v0 = v ↼ (xy), v ↼ 1 = r(v−1 , 1)v0 = ε(v−1 )v0 = v. (2) The map ↼: V ⊗ O(G) → V is a morphism in

H

M:

(v ↼ x)−1 ⊗ (v ↼ x)0 = r(v−2 , x)v−1 ⊗ v0 , v−1 x1 ⊗ v0 ↼ x2 = v−2 x1 ⊗ r(v−1 , x2 )v0 = r(v−1 , x2 )v−2 x1 ⊗ v0 = r(v−2 , x1 )x2 v−1 ⊗ v0 = r(v−2 , x1 )ε(x2 )v−1 ⊗ v0 = r(v−2 , x)v−1 ⊗ v0 .


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(3) The maps ↼ and ⇀ commute: (x ⇀ v) ↼ y = r((x ⇀ v)−1 , y1 )y2 ⇀ (x ⇀ v)0 = r(x1 v−1 , y1 )y2 ⇀ x2 ⇀ v0 = ε(x1 )r(v−1 , y1 )y2 ⇀ x2 ⇀ v0 = r(v−1 , y1 )y2 ⇀ x ⇀ v0 = x ⇀ (v ↼ y). (4) Let f : V → W a morphism in H M(G), to see that f : V → W is a morphism in H G MG is enough to prove that f is a right O(G)-module morphism, f (v ↼ x) = f (r(v1 , x1 )x2 ⇀ v0 ) = r(v1 , x1 )x2 ⇀ f (v0 ) = r(f (v)−1 , x1 )x2 ⇀ f (v)0 = f (v) ↼ x. for all x ∈ O(G), v ∈ V . Therefore we have a well-defined fully faithful functor from H M(G) to Let V, W ∈ H M(G), v ∈ V, w ∈ W, x ∈ O(G), the calculation

H G MG .

r((v ⊗O(G) w)−1 , x1 )x2 ⇀ (v ⊗O(G) w)0 = r(v−1 w−1 , x1 )(x2 ⇀ v0 ) ⊗O(G) w0 = r(w−1 , x1 )r(v−1 , x2 )(x3 ⇀ v0 ) ⊗O(G) w0 = r(w−1 , x1 )(v0 ↼ x2 ) ⊗O(G) w0 = r(w−1 , x1 )v0 ⊗O(G) x2 ⇀ w0 = v ⊗O(G) (w ↼ x). proves that the right O(G)-action of V ⊗O(G) W is induced by the left O(G)-action; in other words, ⊗O(G) defines a monoidal structure on H M(G) such that H M(G) is a tensor subcategory of H G MG . Definition 2.7. Let H be a Hopf algebra and G an affine group scheme, with a central inclusion of G in H. The category H M(G) with the monoidal structure ⊗O(G) is called the de-equivariantization of H by G. 2.3. Tannakian reconstruction of H M(G). Let H be a Hopf algebra and O(G) ⊂ H be a braided central inclusion of G in H. We shall say that the central inclusion of G in H is cleft if there exists a convolution invertible O(G)-linear map π : H → O(G); such a map is called a cointegral. Lemma 2.3 implies that r : H ⊗ O(G) → k induces a well-defined map H/O(G)+ H ⊗ O(G) → k,

h ⊗ k 7→ r(h, k), +

which we will denote again by r (here, O(G) = ker(ε) is the augmentation ideal). The goal of this section is to prove the following result. Theorem 2.8. Let H be a Hopf algebra and (O(G), r) a cleft braided central Hopf subalgebra with cointegral π such that επ = ε and π(1) = 1. Then the quotient coalgebra Q := H/O(G)+ H is a coquasi-bialgebra with multiplication and associator given by: (2.7) (2.8)

m(a ⊗ b) = j(a1 )j(b)1 r(a2 , π(j(b)2 )) ω(a ⊗ b ⊗ c) = r(a, π(j(b)1 j(c)1 ))r(j(b)2 , π(j(c)2 )),

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where a, b, c ∈ Q and j : Q → H, q 7→ π −1 (q1 )q2 . There is a monoidal equivalence between H M(G) and Q M. Before to give the proof of the Theorem, we want to explain briefly the Tannakian reconstruction principle that we shall use. Let C be a coalgebra and C M be the category of left C-comodules. Assume that C M has a monoidal structure ⊗ : C M × C M → C M, αV,W,Z : (V ⊗W )⊗Z → V ⊗(W ⊗Z), 1⊗V = V ⊗1 such that the underlying functor C M → Vectk is a strict quasi-monoidal functor, i.e., V ⊗W = V ⊗k W and 1 = k as vector spaces, then C has a coquasi-bialgebra structure (m, ω) given by (2.9)

m(a, b) = (a ⊗ b)−1 ε((a ⊗ b)0 ),

(2.10)

ω(a, b, c) = ε(α(a ⊗ b ⊗ c))

and the monoidal structure on C M defined by the coquasi-bialgebra structure coincides with the monoidal structure (⊗, α, 1). Proof. From now on we fix a cointegral π such that επ = ε and π(1) = 1. Let Q = H/O(G)+ H be the quotient coalgebra of H, then by [DMR, Theorem 2.4] and [Sch, Theorem II] the functors (2.11) (2.12)

b: V

(2.13)

H

M(G) → Q M M 7→ M = M/K + M

HQ V ←7 V,

+ define a category equivalence, where M P = M/O(G) Q-comodule with P M is a left P m−1 ⊗ m0 = m−1 ⊗ m0 , and HQ V = { h ⊗ v| h1 ⊗ h2 ⊗ v = h ⊗ v−1 ⊗ v0 } ∈ H M(G) has as left H-comodule and a left O(G)-module structures the ones induced by the left tensor factor. By [S1, Lemma 3.3.5], for all M, N ∈ H M(G) we have a linear isomorphism π ξM,N : M ⊗ N → M ⊗O(G) N

m ⊗ n 7→ m ⊗O(G) π −1 (n−1 )n0 r(m−1 , π(n−1 )1 )π(n−1 )2 m0 ⊗ n0 = mπ(n−1 ) ⊗ n0 ←7 m ⊗O(G) n such that the functor (V, ξ π ) : H M(G) → Veck , M 7→ M := M/K + M is quasi-tensor. Then using the equivalence (2.11), the category Q M has a (unique) b is a monoidal equivalence and the following diagram monoidal structure such the V of functors commutes H

M(G) ◗ s ◗ (V,ξ) ◗

b V

✲

Q

M

✑ ✑U ✰ ✑ Veck

Consequently the underlying functor U becomes an strict quasi-monoidal functor and we can apply Tannakian reconstruction. A natural section j : Q → H for the canonical projection νH : H → Q is given by (2.14)

j(h) = π −1 (h−1 )h0 .

Fix a cointegral π : H → O(G) such that π(1) = 1, and define j as in (2.14).


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The Q-comodule structure on Q ⊗ Q is:

(m ⊗ n)−1 ⊗ (m ⊗ n)0 = ξ(m ⊗ n)−1 ⊗ ξ −1 (ξ(m ⊗ n)0 ) = (m ⊗ j(n))−1 ⊗ ξ −1 (m ⊗ j(n)) = m−1 j(n)−1 ⊗ ξ −1 (m0 ⊗ j(n)0 ) = m−1 j(n)−1 ⊗ r(m0,−1 , π(j(n)0,−1 )1 ) π(j(n)0,−1 )2 m00 ⊗ j(n)00 = m−2 j(n)−2 ⊗ r(m−1 , π(j(n)−1 )1 ) π(j(n)−1 )2 m0 ⊗ j(n)0 = j(m−2 )j(n)−2 ⊗ r(m−1 , π(j(n)−1 )1 ) π(j(n)−1 )2 m0 ⊗ j(n)0 ,

for all m, n ∈ H, m, n ∈ Q. Now, applying the formula (2.9), we have

m(m ⊗ n) = j(m−2 )j(n)−2 r(m−1 , π(j(n)−1 )1 )ε(π(j(n)−1 )2 m0 )ε(j(n)0 ) = j(m−2 )j(n)−1 r(m−1 , π(j(n)0 )1 )ε(π(j(n)0 )2 m0 ) = j(m−1 )j(n)−1 r(m0 , π(j(n)0 )),

that is

m(a ⊗ b) = j(a1 )j(b)1 r(a2 , π(j(b)2 )).

The constraint of associativity of diagram

L⊗M ⊗N ξL,M ⊗id

Q

M, is defined by the commutativity of the

αL,M ,N

✲ L⊗M ⊗N

id ⊗ξM,N ❄ ❄ L ⊗O(G) M ⊗ N L ⊗ M ⊗O(G) N ✟ ❍ ❍❍ ✟✟ ξL⊗O(G) M,N ❍❍ ✟✟ ξL,M ⊗O(G) N ❍ ✙ ✟ ❥ L ⊗O(G) M ⊗O(G) N

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Hence, −1 −1 ◦ ξL,M⊗ ◦ ξL⊗O(G) M,N ◦ ξL,M ⊗ id (l ⊗ m ⊗ n) α(l ⊗ m ⊗ n) = id ⊗ ξM,N O(G) N −1 −1 = id ⊗ ξM,N ◦ ξL,M⊗ (l ⊗ j(m) ⊗ j(n)) O(G) N −1 = id ⊗ ξM,N (r(l−1 , π((j(m) ⊗ j(n))−1 )1 )

π((j(m) ⊗ j(n))−1 )2 l0 ⊗ j(m)0 ⊗ j(n)0 ) = r(l−1 , π(j(m)−1 j(n)−1 )1 )id ⊗ −1 ξM,N (π(j(m)−1 j(n)−1 )2 l0 ⊗ j(m)0 ⊗ j(n)0 )

= r(l−1 , π(j(m)−1 j(n)−1 )1 )r(j(m)0,−1 , π(j(n)0,−1 )1 ) π(j(m)−1 j(n)−1 )2 l0 ⊗ π(j(n)0,−1 )2 j(m)00 ⊗ j(n)00 = r(l−1 , π(j(m)−2 j(n)−2 )1 )r(j(m)−1 , π(j(n)−1 )1 ) π(j(m)−1 j(n)−1 )2 l0 ⊗ π(j(n)−1 )2 j(m)0 ⊗ j(n)0 for all l ∈ L, m ∈ M, n ∈ N . Applying the formula (2.10), ω(l ⊗ m ⊗ n) = r(l−1 , π(j(m)−2 j(n)−2 )1 )r(j(m)−1 , π(j(n)−1 )1 ) ε(π(j(m)−1 j(n)−1 )2 l0 )ε(π(j(n)−1 )2 j(m)0 )ε(j(n)0 ) = r(l, π(j(m)−1 j(n)−1 ))r(j(m)0 , π(j(n)0 )), that is ω(a ⊗ b ⊗ c) = r(a, π(j(b)1 j(c)1 ))r(j(b)2 , π(j(c)2 )) for all a, b, c ∈ Q. ′

−1

Remark 2.9. (1) If π : H → K is any integral then π (h) := π(h1 )επ (h2 ) is again an integral such that επ ′ = ε. (2) If π : H → K is an integral, then π(1) ∈ H × , and π ′ (h) := π(h)/π(1) is again an integral such that π ′ (1) = 1. (3) If H is finite dimensional Hopf algebra H, every Hopf subalgebra K ⊂ H admits an integral π : H → K. Proposition 2.10. If H is finite dimensional and G is a constant finite algebraic group, then the coquasi-bialgebra Q defined in Theorem 2.8 admits a coquasi-Hopf algebra structure. Proof. Since G is a constant finite group it follows by [DGNO, Theorem 4.18] that the de-equivariantization is a rigid monoidal category. Since Q is a quotient of H, Q is a finite dimensional and by [S2, Theorem 3.1], Q is a coquasi-Hopf algebra. 3. Applications In the last part of this work we will apply the results of the Section 2 to some particular cases. First we consider the category of G-graded vector spaces, for some group G. Second, we look at quotient of Hopf algebras by central Hopf subalgebras, and view them as a de-equivariantization. Finally we study a family of pointed finite-dimensional coquasi-Hopf algebras, whose dual algebras are a generalization of the quasi-Hopf algebras A(H, s) in [A1].


11

3.1. Baby example. Let Γ be a discrete group, G ⊂ Z(Γ) a central subgroup of Γ, and r : Γ × G → k∗ a bicharacter such that r|G×G = 1. Then the pair (kG, r) is a braided central Hopf subalgebra of kΓ. We shall fix a set of representatives of the right cosets of G in Γ, Q ⊂ G. Thus every element γ ∈ Γ has a unique factorization γ = gq, g ∈ G, q ∈ Q. We assume e ∈ Q. The uniqueness of the factorization Γ = GQ implies that there are well defined maps · : Q × Q → Q, θ : Q × Q → G, determined by the conditions pq = θ(p, q)p · q,

p, q ∈ Q.

The map θ is a 2-cocycle θ ∈ Z 2 (Γ/G, G) where Γ/G acts trivially over G, since G is a central subgroup of Γ. We define a map π : Γ → G, γ 7→ x, where x ∈ G is the unique element such that γ = xp with p ∈ Q, and j : Q → Γ is the inclusion. Now by Theorem 2.8, the de-equivariantization is defined as follows. Let K be the quotient group Γ/G, then the group algebra kK with the 3-cocycle ω(u, v, w) = r(u, θ(v, w)) b the deis a coquasi-Hopf algebra and kK M is tensor equivalent to kΓ M(G), kΓ b equivariantization of M by the affine group scheme G(−) = Alg(kG, −). Now, we will explain how this construction determines the same data of [A1, Example 2.2.6]. If Γ is abelian, the map r : Γ × G → k∗ defines a group morphism b x 7→ r(−, x) such that hT (x′ ), xi = r(x, x′ ) = 1, for all x, x′ ∈ G, thus T : G → Γ, it defines an inclusion of VecG as a Tannakian subcategory of Z(VecΓ ), and the 3-cocycle over K is: ω(u, v, w) = r(u, θ(v, w)) = hT (θ(v, w)), ui. 3.2. Second example: Central extension of Hopf algebras. Let H be a Hopf algebra and (K, r) a braided central Hopf subalgebra, if r(h, x) = ε(hx) for all h ∈ H, x ∈ K, then K ⊂ H is a central Hopf subalgebra and this defines a central inclusion of the group scheme G = Spec(K) in H. Also since K is central, K + H is a Hopf ideal and Q = H/K + H is a quotient Hopf algebra of H. Proposition 3.1. Let H be a Hopf algebra and K ⊂ H a cleft central Hopf subalgebra, then the de-equivariantization of H M by G = Spec(K) is tensor equivalent to the tensor category of comodules over the Hopf algebra Q = H/K + H. Proof. The central Hopf subalgebra K is braided central with r(h, k) = ε(hk) for all h ∈ H, k ∈ K. Then the product and coassociator in the coquasi-bialgebra defined in Theorem 2.8 are m(a ⊗ b) = j(a1 )j(b)1 r(a2 , π(j(b)2 )) = j(a1 )j(b)1 ε(a2 )ε(π(j(b)2 )) = j(a)j(b) = j(a)j(b) = ab, ω(a ⊗ b ⊗ c) = r(a, π(j(b)1 j(c)1 ))r(j(b)2 , π(j(c)2 )) = ε(a)ε(π(j(b)1 j(c)1 ))ε(j(b)2 )ε(π(j(c)2 )) = ε(abc),

12


for all a, b, c ∈ H, a, b, c ∈ Q. Then the coquasi-bialgebra structure is the Hopf algebra quotient structure, and the Q M is tensor equivalent to the de-equivariantization by Spec(K). The interesting point of the Proposition above is that this provides a categorical interpretation of the tensor category Q M in terms of de-equivariantization of an affine group scheme. Example 3.2. Let G be a connected, simply connected complex simple Lie group, and let g be its associated Lie algebra. In [ArG] the authors consider the following setting: an injective map of Hopf algebras ι : O(G) ֒→ A, and a surjective map, π : A → Q, satisfying the conditions i) π ◦ ι(a) = ǫ(a)1Q , for all a ∈ O(G); ii) Acoπ = O(G); iii) ker π = O(G)+ A; iv) either A is flat as O(G)-module, or the functor Ind : Q M → A M is exact and faithful. Therefore, they obtain an equivalence between the category Q M and the deequivariantization of A M by G, see [ArG, Thm. 2.8]. Now, our results give an alternative proof to this equivalence and we can state that this is a tensor equivalence. They apply the result to the following case. Let l ≥ 3 be an odd integer, relative prime to 3 if g contains a G2 -component, and let ζ be a complex primitive l-th root of 1. By Oζ (G) we denote the complex form of the quantized coordinate algebra of G at ζ and by uζ (g) the Frobenius-Lusztig kernel of g at ζ, see [DL] for definitions. We need the following facts about Oζ (G), see [DL, Prop. 6.4]: it fits into the following cocleft central exact sequence 1 → O(G) → Oζ (G) → uζ (g)∗ → 1. Then the tensor category of modules over the Frobenius-Lusztig kernel is a deequivariantization of Oζ (G), which is the main result of [ArG]. Moreover, the main result in [AnG] establishes that any quantum subgroup is obtained as a cocleft central exact sequence, similar to the previous one, that is, we can view these constructions as de-equivariantizations. The same construction works for the restricted two parameter (pointed) quantum group u bα,β (gln ) with the algebraic group GLn , where α, β ∈ k are such that αβ −1 is a root of unity of order l, and αl = β l = 1. According to [Ga, Cor. 5.3, 5.15], we have a central extension of Hopf algebras 1 → O(GLn ) → Oα,β (GLn ) → u bα,β (gln )∗ → 1.

Therefore Proposition 3.1 shows that the category of modules over u bα,β (gln ) is the de-equivariantization of the category of comodules over Oα,β (GLn ) by GLn . A similar situation holds for any quantum subgroup of this quantum group. 3.3. A generalization of the family of algebras A(H, s). In this Subsection we shall assume that k is an algebraically closed field of characteristic zero. b = Hom(Γ, k∗ ). We consider a finite-dimensional Let Γ be a finite group and Γ coradically graded pointed Hopf algebra H = ⊕n≥0 Hn , with G(H) = Γ. We assume that H is generated as an algebra by Γ and H1 ; this is always the case if Γ is abelian, see [A2, Theorem 4.15]. We fix a basis x1 , · · · , xθ of the space V of coinvariants


13

of H1 , so H ≃ B(V )#kΓ, where B(V ) is the Nichols algebra associated to V , and ∆(xi ) = xi ⊗ gi + 1 ⊗ xi for some gi ∈ Γ. L Proposition 3.3. Let H = ∞ n=0 Hn , Γ, x1 , · · · , xθ be as before. There exists a bijection between (a) central braided Hopf subalgebras (K, r), and b is a (b) pairs (G, Φ), where G is a central subgroup of Γ, and Φ : G → Γ morphism of group such that hg ′ , Φ(g)i = 1,

gxi g −1 = hgi , Φ(g)ixi ,

for all g, g ′ ∈ G, 1 ≤ i ≤ θ. The correspondence is given by defining K = kG, and extending the evaluation map < ·, Φ(·) >: Γ × G → k, linearly to H0 ⊗ K, and as zero over Hn , n ≥ 1. Proof. Given a central braided Hopf subalgebra K ⊂ H, we have that K ⊂ H0 is commutative, so K = kG for some subgroup G of Γ. By (2.4), G is inside the b given by center of Γ. By (2.1) and (2.2), we have a morphism of groups Φ : G → Γ < γ, Φ(g) >:= r(γ, g),

γ ∈ Γ, g ∈ G,

such that hg ′ , Φ(g)i = 1 for all g, g ′ ∈ G. Now by (2.2) we have also that r(xi , g) ggi + gxi = r(gi , g) xi g + r(xi , g) g, so r(xi , g) = 0, and gxi g −1 = r(gi , g)xi for all i and all g ∈ G, because H is graded and gi 6= 1. As H is generated by skew primitive and group-like elements, we deduce that r(x, k) = 0, for all k ∈ K, x ∈ Hn , n ≥ 1. The converse is easy to prove.

Remark 3.4. Fix a set Q ⊂ Γ of representatives of the right cosets of G in Γ. Note that the map π : H → K = kG given as in Subsection 3.1 over H0 , and extended as 0 over the other components, is an integral for K. L Definition 3.5. Let H = n≥0 Hn be a coradically graded finite-dimensional Hopf algebra such that H0 = kΓ, where Γ is a finite group, and H is generated by group-like and skew-primitive elements. For each pair (G, Φ) as in the Proposition 3.3, we shall denote by A(H, G, Φ) the coquasi-Hopf algebra associated, constructed by using Theorem 2.8. Example 3.6. Let H = ⊕n≥0 Hn be as above, where G(H) = Γ is an abelian group. Therefore H is generated by the group-like elements and a finite set x1 , . . . , xθ of (γi , 1)-primitive elements, i.e. ∆(xi ) = xi ⊗ γi + 1 ⊗ xi ,

i ∈ {1, . . . , θ},

b such that and also we can suppose that there are characters χi ∈ Γ γxi γ −1 = χi (γ)xi ,

i ∈ {1, . . . , θ}, γ ∈ Γ.

In this case, Φ satisfies the condition χi (γ) = hγi , Φ(γ)i for all i ∈ {1, . . . , θ} and all γ ∈ Γ. Therefore Φ(γ) is uniquely determined (and possibly it does not exist) when the γi ’s generate Γ as a group.

14


The Nichols algebra B(V ) admits a Nθ -gradation, and we can fix a basis B of B(V ) whose elements are Nθ -homogeneous, and such that 1 ∈ B. For each x ∈ B we denote |x| ∈ Nθ its degree, and γx := γ1a1 · · · γθaθ ,

χx := χa1 1 · · · χaθ θ ,

if |x| = (a1 , . . . , aθ ) ∈ Nθ .

Therefore γxγ −1 = χx (γ)x for all γ ∈ Γ, and ∆(x) is written as the sum of x ⊗ 1 plus γx ⊗ x plus terms in intermediate degrees for the N-gradation. Fix a set of representatives elements q1 = e, . . . , qt ∈ Γ of Γ/G, so Q has a basis (qi x)x∈B,1≤i≤t . Therefore the multiplication and the associator of Q are given by: m(qi x, qj y) = r qi γx , π(qj γy ) qi x · qj y = Φ π(qj γy ) (qi γx )qi x · qj y, ω(qi x, qj y, qk z) = r qi , π(qj qk ) r qj , π(qk ) δx,1 δy,1 δz,1 for any x, y, z ∈ B and 1 ≤ i, j, k ≤ t.

More concretely, suppose that Γ is a cyclic group of order m2 , generated by γ, and that x1 , . . . , xθ are the skew-primitive elements. Thus, if q is a primitive m2 -roof of unity, there are unique integers di , bi , module m2 , such that χi (γ) = q di ,

∆(xi ) = xi ⊗ γ bi + 1 ⊗ xi .

b such that χ(γ) = q. A morphism Set G = hgi, where g = γ n , so G ≃ Zn , and χ ∈ Γ b Φ : G → Γ is determined by an integer s (unique modulo n) such that Φ(g) = χns . Therefore the conditions in Proposition 3.3 are satisfied for each element in Υ′ (H) := {s : 0 ≤ s ≤ n − 1,

bi s ≡ di (n), ∀i = 1, . . . , θ}.

A set of representatives of Γ/G ≃ Zn is given by γ i , 0 ≤ i ≤ n − 1. Remark 3.7. If H is a finite dimensional Hopf algebra as in this example, then H ∗ also is of this type and Υ′ (H ∗ ) = Υ(H), where Υ(H) was defined in [A1]. For each s ∈ Υ′ (H) there exists a coquasi-Hopf algebra A′ (H, s). We identify the group of simple (one-dimensional) comodules with Zn , and the 3-cocycle determining the associator is ′ 0 ≤ i, j, k ≤ n − 1, ω(γ i , γ j , γ k ) = q nsi j+k−(j+k) , where j ′ denotes the remainder of j in the division by n. Note that A′ (H, s) is dual to the quasi-Hopf algebra A(H ∗ , s) of [A1], and these quasi-Hopf algebras include the examples in [Ge]. Example 3.8. We consider now de-equivariantizations of some pointed Hopf algebras related with small quantum groups by applying the previous construction. We fix then a finite Cartan matrix A = (aij )1≤i,j≤θ corresponding to a semisimple Lie algebra g, positive integers di , 1 ≤ i ≤ θ such that they are the minimal ones satisfying di aij = dj aji , and let ∆+ be its set of positive roots and M := |∆+ |. Fix also a root of unity q of order N = mn, m, n > 1, qij := q di aij , {αi } the canonical basis of Zθ , χ : Zθ × Zθ → k× the bicharacter determined by χ(αi , αj ) = qij , 1 ≤ i, j ≤ θ. Let qβ := χ(β, β) and Nβ := ord qβ , for each β ∈ ∆+ . We will describe the corresponding Nichols algebra B(V ) of diagonal type attached to (qij ) and the corresponding Hopf algebra obtained by bosonization by a particular abelian group. We refer to [A2, Theorems 1.25, 3.1] for the corresponding statements about the Nichols algebra. Fix a basis x1 , . . . , xθ of V , the group Γ = (ZN )θ , with generators γ1 , . . . , γθ of each cyclic group of order N , and


15

consider the realization of V as a Yetter-Drinfeld module with comodule structure determined by δ(xi ) = γi ⊗ xi . Recall that the braided adjoint action of xi has the following property: y ∈ B(V ) Zθ − homogeneous of degree β.

(ad c xi )y := xi y − χ(αi , β)yxi ,

The associated finite-dimensional pointed Hopf algebra H = B(V )#kΓ is described as follows. As an algebra, it is generated by γ1 , . . . , γθ , x1 , . . . , xθ , which satisfies the following relations: γiN = 1,

γi γj = γj γi ,

(ad c xi )1−aij xj = 0,

γi xj = qij xj γi , N

xβ β = 0,

i 6= j,

β ∈ ∆+ ,

if N ≥ 8 (otherwise we need extra relations). Each xβ is an homogeneous element of B(V ) of degree β, obtained for a fixed convex order on the roots β1 < β2 < · · · < βM , and H has a PBW basis B as follows: n o b1 γ1a1 · · · γθaθ xbβM . · · · x : 0 ≤ a < N, 0 ≤ b < N i j β j β1 M The coproduct is determined by ∆(γi ) = γi ⊗ γi ,

∆(xi ) = xi ⊗ γi + 1 ⊗ xi .

We consider ni , mi ∈ N such that N = ni mi for each 1 ≤ i ≤ θ. For each a ∈ N, we denote by µi (a) the remainder of a on the division by ni . Call gi = γini , and let G be the subgroup of Γ generated by g1 , . . . , gθ . Therefore, G′ := Γ/G ≃ Zn1 × · · · × Znθ ,

G ≃ Zm1 × · · · × Zmθ ,

and a set of representatives of G′ is given by γ1a1 γ2a2 , 0 ≤ ai < ni . With this information we can determine j : Q → H and π : H → kG by µ (a1 )

j(γ1a1 · · · γθaθ x) = γ1 1 π(γ1a1

· · · γθaθ x)

=

µ (aθ )

· · · γθ θ

a −µ (a ) γ1 1 1 1

x,

a −µ (a ) · · · γθ θ θ θ ε(x),

where ai ∈ N, x = xbβM · · · xbβ11 . By Proposition 3.3 we have that hγj , Φ(gi )i = M ni χj (gi ) = qij for each pair 1 ≤ i, j ≤ θ, so Φ is univocally determined, and we need the extra conditions N |ni nj , which is equivalent to mi |nj , because hgj , Φ(gi )i = 1 for all i, j. To determine explicitly the coquasi-Hopf algebra structure of Q, we consider the basis o n B := γ1a1 · · · γθaθ x : 0 ≤ ai < ni , x = xbβM · · · xbβ11 . M Given two elements x, y of B(V ) of degree (e1 , . . . , eθ ), (f1 , . . . , fθ ) ∈ Nθ0 , respectively, and 0 ≤ ai , bi < ni , we compute Y (b +f −µ (b +f ))(a +e )−b e qij j j j j j i i j i m γ1a1 · · · γθaθ x, γ1b1 · · · γθbθ y = 1≤i,j≤θ

µ (a +b ) γ1 1 1 1

µ (aθ +bθ )

· · · γθ θ

xy,

For each x, y, z ∈ B(V ), 0 ≤ ai , bi , ci < ni , the associator is computed as Y (b +c −µ (b +c ))a qij j j j j j i . ω γ1a1 · · · γθaθ x, γ1b1 · · · γθbθ y, γ1c1 · · · γθcθ z = δx,1 δy,1 δz,1 1≤i,j≤θ

16


Note that we can obtain the quasi-Hopf algebras appearing in [EG] as the dual structures of the coquasi-Hopf algebras obtained for Γ = (Zn2 )θ and G ∼ = (Zn )θ as 2 a subgroup of Γ, i.e. N = n , mi = ni = n. Example 3.9. Finally we consider some de-equivariantizations related with a Nichols algebra of diagonal type but not of Cartan type. Consider a braiding whose diagram is the last one of row 9 in [H, Table 1], and an associated H = B(V )#kΓ. Here we fix Γ = Z9N × Z18M , with generators γ1 , γ2 of each cyclic subgroup, respectively, and a root of unity q of order 9. Using the presentation given for the corresponding Nichols algebra in [A2, Theorem 3.1], we can describe H as follows. As an algebra, it is generated by γ1 , γ2 , x1 , x2 , and relations γ19N = γ218M = 1,

γ1 γ2 = γ2 γ1 ,

γi xj γi−1 = qij xj ,

18 x31 = x22 = x18 12 = x112 = x112 y − qyx112 = 0,

where q11 = q 3 , q12 = q 4 = q21 , q11 = −1, (ad c xi )y := xi y − (γi yγi−1 )x for each i = 1, 2, and each y ∈ B(V ), and we consider: x12 = (ad c x1 )x2 ,

y = x112 x12 + x12 x112 ,

2

z = yx12 − q 2 x12 y,

x112 = (ad c x1 ) x2 ,

so by [A2, Theorem 1.25] H has a PBW basis B as follows: n 2 γ1a1 γ2a2 xb26 xb125 z b4 y b3 xb112 xb11 : 0 ≤ a1 < 9N, 0 ≤ a2 < 18M, 0 ≤ b2 , b5 < 18, b1 , b3 ∈ {0, 1, 2}, b4, b6 ∈ {0, 1}, } . The coproduct is determined by ∆(γi ) = γi ⊗ γi ,

∆(xi ) = xi ⊗ γi + 1 ⊗ xi .

Fix n, n1 , m, m1 ∈ N such that 9N = nn1 and 18M = mm1 . For each a ∈ N, we denote by a′ (respectively, a′′ ) the remainder on the division by n (respectively, m). Call g1 = γ1n , g2 = γ2m , and G the subgroup of Γ generated by g1 and g2 . Therefore, G ≃ Zn1 × Zm1 ,

G′ := Γ/G ≃ Zn × Zm ,

and a set of representatives of G′ are γ1a1 γ2a2 , 0 ≤ a1 < n, 0 ≤ a2 < m. Also, we can write explicitly: a′

a′′

j(γ1a1 γ2a2 x) = γ1 1 γ2 2 x,

a −a′1 a2 −a′′ 2 γ2 ε(x)

π(γ1a1 γ2a2 x) = γ1 1

∈ G,

ai ∈ N, x ∈ B.

By Proposition 3.3 we have that n hγ1 , Φ(g1 )i = χ1 (g1 ) = q11 ,

n hγ2 , Φ(g1 )i = χ2 (g1 ) = q12 ,

m hγ1 , Φ(g2 )i = χ1 (g2 ) = q21 ,

m hγ2 , Φ(g2 )i = χ2 (g2 ) = q22 ,

so Φ is univocally determined, and moreover it tells us that m, n should satisfy 3|n, 2|m, 9|mn, because hgj , Φ(gi )i = 1 for all i, j ∈ {1, 2}. We compute the structure of the coquasi-Hopf algebra associated to this datum. Note that the following set is a basis of Q: n o B := γ1a1 γ2a2 x : 0 ≤ a1 < n, 0 ≤ a2 < m, x ∈ B .


17

Given x, y ∈ B of degree (e1 , e2 ), (f1 , f2 ) ∈ N20 , respectively, and 0 ≤ a1 , b1 < n, 0 ≤ a2 , b2 < m, we have that (b +f1 −(b1 +f1 )′ )(a2 +e2 )−b1 e2

−b1 e1 q121 m(γ1a1 γ2a2 x, γ1b1 γ2b2 y) = q11

(b +f2 −(b2 +f2 )′′ )(a1 +e1 )−b2 e1 −b2 e2 (a1 +b1 )′ (a2 +b2 )′′ γ2 xy, q22 γ1

q212

where we use that 3|n, 2|m. And the associator is given by a (b1 +c1 −(b1 +c1 )′ ) a1 (b2 +c2 −(b2 +c2 )′′ ) q21 ,

ω(γ1a1 γ2a2 x, γ1b1 γ2b2 y, γ1c1 γ2c2 z) = δx,1 δy,1 δz,1 q122

where x, y, z ∈ B, 0 ≤ a1 , b1 , c1 < n, 0 ≤ a2 , b2 , c2 < m. References [AnG]

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´ tica y Estad´ıstica Rafael Laguardia. Facultad de IngeM. P.: Instituto de Matema ´ blica. J.H.Reissig 565, CP 11.300 , Montevideo, Uruguay. nier´ıa. Universidad de la Repu E-mail address: [email protected]